Dynkin index

In mathematics, the Dynkin index

of finite-dimensional highest-weight representations of a compact simple Lie algebra

relates their trace forms via

is the highest root, so that

is the adjoint representation, the Dynkin index

is equal to the dual Coxeter number.

is the trace form on the representation

By Schur's lemma, since the trace forms are all invariant forms, they are related by constants, so the index is well-defined.

Since the trace forms are bilinear forms, we can take traces to obtain[citation needed] where the Weyl vector is equal to half of the sum of all the positive roots of

is the value of the quadratic Casimir in the representation